All Questions
Tagged with metric-tensor lagrangian-formalism
187 questions
-2
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1
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68
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Divergence of canonical energy-momentum tensor in QFT [closed]
I have to show that the divergence of the canonical energy-momentum tensor is zero, i.e. $$\partial^{\mu}T_{\mu\nu} = 0.$$
The Lagrangian is $$\mathcal{L} = \frac{1}{2}\partial^{\mu} \phi \partial_{\...
- 141
2
votes
3
answers
82
views
Deriving differential equation for the path of a particle in potential $U(r)$ using Maupertuis’ principle
I came across this Maupertuis' principle in Landau and Lifshitz, which, in it's final form looks like $$\delta\int\sqrt{2m(E-U)}dl=0.\tag{44.10}$$
They used this equation to show that path of a free ...
- 109
0
votes
2
answers
85
views
Lorentz scalar Lagrangian in curved spacetime
This question might be very simple but I guess I'm missing something. We know that Lagrangian has to be a Lorentz scalar. I can see why that should be the case when dealing with inertial frames of ...
- 387
3
votes
2
answers
354
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Is proper time still maximized with different metric signature?
A free particle moving from event $a$ to event $b$, which is timelike connected to $a$, on spacetime follows geodesics. In most cases it is a path that minimizes this integral
$$S=\int_a^b ds=\int_a^b ...
- 711
1
vote
0
answers
97
views
A confusing sign in Maxwell's equations with differential forms
I'm running into a basic issue with Maxwell's equations in terms of differential forms. Let's work in Minkowski space with mostly negative signature, and define the Hodge star in the usual way,
$$\...
- 105k
1
vote
0
answers
40
views
On the Gauge-fixing for the case of the Polyakov string action
In the book "String theory and M-theory" by Becker-Becker-Schwarz, the author says that
"reparametrization invariance of the string sigma-model action $$S_{\sigma}=\frac{-T}{2}\int d^2 ...
- 634
0
votes
0
answers
45
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Equation of motion for $X^{\mu}$ (geodesic equation)
The action for a relativistic particle of mass $m$ in a curved $D$-dimensional is $$\tilde{S}_0=\frac{1}{2}\int d\tau (\dot{X}^2-m^2)$$ for particular gauge and $\dot{X}^2=g_{\mu\nu}(X)\dot{X}^{\mu}\...
- 634
3
votes
1
answer
94
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Where this definition $T_{\alpha\beta}=-\frac{2}{T}\frac{1}{\sqrt{-h}}\frac{\delta S}{\delta h^{\alpha \beta}}$ come from?
In the book "String theory and M-theory" by Becker, Becker and Schwarz, the author says that the Nambu-Goto action $$S_{NG}=-T\int d\sigma\, \tau \sqrt{(\dot{X}\cdot X')^2 -\dot{X}^2X'^2}$$ ...
- 634
1
vote
1
answer
52
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How to calculate the functional derivative of a product?
Let $\omega_{J}^{I}=\omega^{I}_{\nu~J}dx^{\nu}$ be a one-form connection with values in the Lorentz group $SO(3,1)$ and $$B_{J}^{I}=B^{I}_{\mu\nu~J}dx^{\mu}\wedge dx^{\nu}$$ a two-form with values in ...
- 176
2
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1
answer
140
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On the Background Independence condition
In General Relativity, one has that the equations of motion for any matter distribution are given by extremizing the following action:
\begin{equation}
S[g] = \int\left[\frac{1}{8\pi}(R - 2\Lambda) + ...
- 562
1
vote
1
answer
67
views
How do you differentiate $F^{αβ}$ with respect to $g_{μν}$?
I want to experiment with this relation (from Dirac's "General Theory of Relativity"):
$$T^{μν} = -\left(2 \frac{∂L}{∂g_{μν}} + g^{μν} L \right)$$
using the electromagnetic Lagrangian $L = -(...
- 467
1
vote
1
answer
113
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Classical open string in Polchinski -- consistency of Neumann boundary conditions with gauge choice
In Section 1.3 of String Theory, Volume 1, Polchinski derives the open string spectrum from the Polyakov action with Neumann boundary conditions, by first considering the classical open string in ...
- 23
3
votes
0
answers
95
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Connection between the metric tensor and mass
The general expression of a line element in a space with metric tensor $g_{\mu \nu}$ is $$ds = \sqrt{ g_{\mu \nu} dX^{\mu} dX^{\nu} }$$
If we consider a curve $X^{\mu}(\tau)$ parametrised by $\tau$, ...
- 347
2
votes
3
answers
153
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Sign conventions for the Lagrangian from the EM Lagrangian density
In Chapter 13.6 of the 3rd edition of Goldstein's Classical Mechanics, Goldstein proposes the Lagrangian density of the electromagnetic field as:
$$\mathcal{L} = -\frac{F_{\lambda \rho} F^{\lambda \...
- 45
0
votes
0
answers
34
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Turning a Lagrangian contains superscript and subscript indices into energy
I'm recently reading the book "Solitons and Instantons" written by R. RAJARAMAN. However, for lacking of ability, I couldn't figure out how to derivate the static solution for energy with ...
- 1
0
votes
0
answers
46
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Variation of action of non-critical string under Weyl transformation (worldsheet cosmological constant term)
In David Tong's lecture notes on string theory, section 5.3.2 An Aside: Non-Critial Strings, page 121, he describes the non-critical string with the following action:
$$S_{\text{non-critical}} = \frac{...
- 327
0
votes
0
answers
33
views
Howe-Tucker to Nambu-Goto Action
Aim to find from the Howe-Tucker action:
$$S_{\text{HT}}=-\frac{1}{2}\int d^d\sigma\sqrt{-\gamma}(\gamma^{ab}\partial_a X^{\mu}\partial_b X^{\nu}\eta_{\mu\nu}-m^2(d-2))$$
(which is a Polyakov-like ...
- 1
4
votes
2
answers
83
views
Functional variation about metric
Recently I’m learning about lagrangian fomulation in GR. And I’m doing some calculations for some toy models.
I precisely understand the variance of christoffel, Riemann tensors about metric, but when ...
- 61
0
votes
1
answer
112
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The Klein-Gordon equation and the sign of the mass term
A derivation of the Klein-Gordon equation starts with the following lagrangian for a scalar field ϕ:
$$
L=\frac{1}{2}g^{ab}(∇_a\phi)(∇_b\phi)-V(\phi)
$$
If we plug this lagrangian in the Euler-...
- 141
2
votes
0
answers
48
views
Sigma-Omega model on curved space-time
I'm trying to get the equations of motion for the scalar meson $\sigma$, vector meson field $\omega_{\mu}$ and finally for the nucleons $\Psi=(\Psi_n,\Psi_p)^{T}$ in the sigma omega model on a curved ...
- 157
4
votes
0
answers
122
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Do functional derivatives commute in a specific example?
Edit: this question is related to other already asked questions, like Symmetry of second functional derivatives , but a clear and definitive answer has never been given, and I am here giving a ...
- 41
7
votes
2
answers
1k
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Why does the Minkowski matrix appear in the free particle action?
It is usual to write the "kinetic" part of the SR action as the Minkowski space-time interval, here $(-,+,+,+)$, times $mc$
$$
S_{kin} = -\int_{\tau_1}^{\tau_2}mc\sqrt{-\eta_{\mu\nu}\dot{x}^{...
- 391
1
vote
1
answer
86
views
Sign convention for the Lagrangian of a free massive point particle in general relativity
As far as I understand, the Lagrangian of a massive free particle in the context of general relativity is the following:
$$L=-mc\sqrt{g_{\mu\nu}\dfrac{dx^\mu}{dt}\dfrac{dx^\nu}{dt}}.$$
But is this the ...
- 323
2
votes
0
answers
239
views
How does one justify Schwartz's answer to his problem 3.7 in "Quantum Field Theory and the Standard Model"?
In problem 3.7 of his textbook "Quantum Field Theory and the Standard Model", Schwartz gives a simplified Lagrangian density
$$
\mathcal{L}=- \frac{1}{2} h \Box h + \epsilon^a h^2 \Box h -\...
- 137
1
vote
0
answers
87
views
How to do taylor expansion of metric tensor to get Lagrangian for first Post-Newtonian order? [closed]
For the two-body compact object system how to write Lagrangian after obtaining first post-Newtonian metric tensor(just gravitational field)? Considering the perturbation in the metric as $$gαβ=ηαβ+hαβ,...
1
vote
0
answers
47
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On the metric signature and the energy-momentum tensor [duplicate]
Given a Lagrangian $$\mathscr{L} = -\frac{1}{2}\partial_\mu \phi \partial^\mu \phi - V(\phi),\tag{1}$$ is the metric always with signature $(-, +, +, +)$? It seems to me that This post Sign Convention ...
- 283
1
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0
answers
30
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Understanding classically equivalent actions of the same physical theory - what went wrong as they produce different E.O.M? [duplicate]
I am working on a specific example where the metric I am using is the $AdS_4$ metric whose ricci scalar $R=-12/l^2$ for some characteristic scale $l$: $$ds^2=-\cosh^2\left(\frac{\rho}{l}\right)dt^2+d\...
- 862
2
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1
answer
106
views
Tensionless string in Nambu-Goto action
I am studying string theory from the book "String theory and M-theory", written by Becker, Becker and Schwartz. My question is:
We are taught that one of the advantages of introducing a ...
- 4,157
2
votes
1
answer
146
views
Nambu-Goto action and the World-Sheet Area
I am studying string theory from the book "String theory and M-theory", written by Becker, Becker and Schwartz. My question is:
We are told that the Nambu-Goto action is simply the one that ...
- 4,157
2
votes
1
answer
63
views
Calculations with co- and contravariant formalism in QFT
i have another question regarding calculations with the co- and contravariant formalism in QFT. It is not that i don't understand all of this, but most of the time i'm missing some "middle" ...
-2
votes
1
answer
147
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Finding equation of motion for given Lagrangian with respect to metric
Given the following action in $d$ dimensional $(0,1,...,d-1)$ curved spacetime:
$$ S= \int d^dx\sqrt{-g}\mathscr{L}[\chi,\Phi,g^{\mu\nu}] $$
Where:
$$\mathscr{L}=e^{-2\Phi} \left(-\frac{1}{2\kappa^2}[...
0
votes
1
answer
599
views
Writing the Euler-Lagrange equation for variation of an action with respect to metric using only the Lagrangian
Given Lagrangian which dependent on collection of fields ${\phi^a},a=1,...,N$
and on a tensor metric $g^{\mu\nu}$ such that the action in $d$ dimension which describes the system is $$S=\int d^dx \...
1
vote
0
answers
87
views
Dirac's pseudo-energy tensor
Having trouble with Dirac's eq. (31.3) ("General Theory of Relativity"). Probably a simple math question, but I need to provide a little background on the symbols.
Dirac defines the pseudo-...
- 467
3
votes
0
answers
193
views
Is there a connection between geodesics and the Euler-Lagrange equation for mechanical systems? [duplicate]
Given an affine manifold $(M,\nabla)$, the geodesic equation $\ddot{x}^j+\dot{x}^k \dot{x}^l\Gamma_{kl}^j=0$ completely characterizes the geodesics on the manifold. This is often called the Euler-...
- 143
0
votes
1
answer
61
views
What does it mean to have a zero-dimensional induced metric?
I have an integral on the form
\begin{equation}
S=\int d^dx g_{\mu \nu} h^{ab}.
\end{equation}
In this example, $g_{ab}$ is a $d$-dimensional metric, $h_{ab}$ is an co-dim 2 induced metric. I wanted ...
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1
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0
answers
165
views
Energy-momentum tensor of Nambo-Goto action?
In all lecture notes on introductory string theory people compute the energy-momentum tensor from the Polyakov action, not the Nambu-Goto action so I thought I would give that a go but I am a bit ...
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3
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0
answers
96
views
What does the Einstein-Hilbert action look like in terms of Riemannian metric of positive signature?
For a 4-manifold to admit a Lorentzian metric is equivalent to that manifold having vanishing Euler characteristic. Any spacetime that admits a Lorentzian metric $g^{\mathcal{L}}$ can have that metric ...
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0
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2
answers
124
views
How to understand the quadratic form of kinetic energy with $\dot{q}$ coefficients?
Kinetic energy can be written as:
$$ T=\frac{1}{2}\sum_{\alpha=1}^K\sum_{\beta=1}^K a_{\alpha \beta}(q)\dot{q}^\alpha \dot{q}^\beta$$
Where the object $a_{\alpha \beta}$ is a certain tensor. How to ...
- 2,180
1
vote
1
answer
103
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String action in light-cone coordinates
I am going through textbook Einstein Gravity in a Nutshell by A. Zee and I got mathematically stuck at page 147 where he is talking about the classical string action using light cone coordinates. ...
1
vote
1
answer
1k
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Ricci scalar variation on conformal transformation and scalar field lagrangian
I'm interested in finding the constraints on constants $p,\xi$ appearing on a scalar field lagrangian density conformally coupled to spacetime. The transformation is
$$ \tilde{g}_{\alpha\beta} = a^2 ...
- 880
0
votes
1
answer
184
views
Problem with linearization of Einstein-Hilbert action in de Sitter background
For some purpose, I have to calculate the well-known linearization of the Einstein-Hilbert action in the de Sitter background. I encountered a problem: assuming the de Sitter metric, my resulting ...
- 2,803
1
vote
1
answer
944
views
Derivation of Einstein-Hilbert Lagrangian [duplicate]
How did Hilbert derive the Lagrangian $\frac{1}{2 \kappa} R \sqrt{-g}$ for the Einstein-Hilbert action?
If it was defined rather than being derived to be that way, then what was the behind it?
How did ...
- 73
5
votes
2
answers
2k
views
Euler-Lagrange equation in curved spacetime
The action of a field $\phi^\mu$ in flat $n$-dimensional spacetime is
$$ S = \int \text{d}^n x \mathscr{L}(\phi^\mu(x),\partial_\alpha \phi^\mu(x)) $$
From an infinitesimal variation of field ...
- 880
2
votes
0
answers
110
views
Variation of normal, coordinate and mean curvature with respect to metric
Using the divergence theorem, we can compute volume by integrating along the surface:
$$\mathrm{vol}(M)=\int_M\mathrm{d}V=\oint_{\partial M}\mathrm{d}S \,\vec{n}\cdot\vec{v},$$
where $\vec{n}$ is ...
- 316
2
votes
1
answer
97
views
Variation of functional with area and volume term
Let $M$ be a closed manifold in $\mathbb{R}^3$ and $\partial M$ its surface. I want to find (in general terms) the manifold that minimizes a functional of the form
$$I[M]=\int_{\partial M}f\,\mathrm{d}...
- 316
0
votes
1
answer
46
views
Asymmetry of the indices in a Weyl transformation for the Polyakov action
The Polyakov action
$${\mathcal {S}}={T \over 2}\int \mathrm {d} ^{2}\sigma {\sqrt {-h}}h^{ab}g_{\mu \nu }(X)\partial _{a}X^{\mu }(\sigma )\partial _{b}X^{\nu }(\sigma )$$
is invariant under the ...
- 1,610
2
votes
1
answer
60
views
Finding the maps inside of the Nambo-Goto action
Let's sps that $\phi:M \to N$ is a diffeomorphism where $M$ and $N$ are two separate manifolds. Let $f$ be a function s.t. $f:M \to R$ where $R$ is the set of all real numbers. The composition of our ...
- 385
3
votes
2
answers
1k
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Derivative of the Lagrangian with respect to the metric tensor
I'm trying to calculate the derivative of the Lagrangian $$\mathcal{L}=\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi-\frac{1}{2}m^2\phi^2$$ with respect to the metric tensor $g_{\mu\nu}$, with the ...
- 154
4
votes
4
answers
809
views
How is proper time extremized?
I just completed an exercise that asked me to prove that, in special relativity, free particles move with uniform velocity on geodesics that are straight lines. After doing this problem, I was ...
- 1,397
1
vote
2
answers
214
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What does general relativity's metric tensor have to do with quantum electrodynamics?
Sabine Hossenfelder recently posted a YouTube video titled, The Closest we have to a Theory of Everything.
At 9:15, she shows the action $S$ for electrodynamics and, immediately after, the Einstein-...
- 4,709